Schur-Weyl duality for tensor powers of the Burau representation

Abstract

Artin's braid group $B_n$ is generated by ${\sigma }_1,\dots ,{\sigma }_{n-1}$ subject to the relations \[ σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1}, \quad σ_iσ_j = σ_j σ_i \text{ if } |i-j|>1. \] For complex parameters $q_1,q_2$ such that $q_1{q}_2\ne 0$, the group $B_n$ acts on the vector space $\mathbf{E}={\sum }_i\mathbb{C}{\mathbf{e}}_i$ with basis ${\mathbf{e}}_1,\dots ,{\mathbf{e}}_n$ by \begin{gather*} σ_i \cdot \mathbf{e}_i = (q_1+q_2)\mathbf{e}_i + q_1\mathbf{e}_{i+1}, \quad σ_i \cdot \mathbf{e}_{i+1} = -q_2\mathbf{e}_i, \\ σ_i \cdot \mathbf{e}_j = q_1 \mathbf{e}_j \text{ if } j \ne i,i+1. \end{gather*} This representation is (a slight generalization of) the Burau representation. If $q=-q_2/q_1$ is not a root of unity, we show that the algebra of all endomorphisms of ${\mathbf{E}}^{\otimes r}$ commuting with the $B_n$-action is generated by the place-permutation action of the symmetric group $S_r$ and the operator $p_1$, given by \[ p_1(\mathbf{e}_{j_1} \otimes \mathbf{e}_{j_2} \otimes \cdots \otimes \mathbf{e}_{j_r}) = q^{j_1-1} \, \sum_{i=1}^n \mathbf{e}_i \otimes \mathbf{e}_{j_2} \otimes \cdots \otimes \mathbf{e}_{j_r} . \] Equivalently, as a $(\mathbb{C}{B}_n,{\mathcal{P}}_r^{\prime }([n{]}_q))$-bimodule, ${\mathbf{E}}^{\otimes r}$ satisfies Schur--Weyl duality, where ${\mathcal{P}}_r^{\prime }([n{]}_q)$ is a certain subalgebra of the partition algebra ${\mathcal{P}}_r([n{]}_q)$ on $2r$ nodes with parameter $[n{]}_q=1+q+\cdots +q^{n-1}$, isomorphic to the semigroup algebra of the "rook monoid" studied by W. D. Munn, L. Solomon, and others.

37 pages